A Decoupled and Positivity-Preserving DDFVS Scheme for Diffusion Problems on Polyhedral Meshes

We propose a decoupled and positivity-preserving discrete duality finitevolume (DDFV) scheme for anisotropic diffusion problems on polyhedral meshes withstar-shaped cells and planar faces. Under the generalized DDFV framework, two setsof finite volume (FV) equations are respectively constructed on the dual and primarymeshes, where the ones on the dual mesh are derived from the ingenious combinationof a geometric relationship with the construction of the cell matrix. The resulting system on the dual mesh is symmetric and positive definite, while the one on the primarymesh possesses an M-matrix structure. To guarantee the positivity of the two categories of unknowns, a cutoff technique is introduced. As for the local conservation, itis conditionally maintained on the dual mesh while strictly preserved on the primarymesh. More interesting is that the FV equations on the dual mesh can be solved independently, so that the two sets of FV equations are decoupled. As a result, no nonlineariteration is required for linear problems and a general nonlinear solver could be usedfor nonlinear problems. In addition, we analyze the well-posedness of numerical solutions for linear problems. The properties of the presented scheme are examined bynumerical experiments. The efficiency of the Newton method is also demonstrated bycomparison with those of the fixed-point iteration method and its Anderson acceleration.     

An Efficient Positivity-Preserving Finite Volume Scheme for the Nonequilibrium Three-Temperature Radiation Diffusion Equations on Polygonal Meshes

This paper develops an efficient positivity-preserving finite volume schemefor the two-dimensional nonequilibrium three-temperature radiation diffusion equations on general polygonal meshes. The scheme is formed as a predictor-corrector algorithm. The corrector phase obtains the cell-centered solutions on the primary mesh,while the predictor phase determines the cell-vertex solutions on the dual mesh independently. Moreover, the flux on the primary edge is approximated with a fixedstencil and the nonnegative cell-vertex solutions are not reconstructed. Theoretically,our scheme does not require any nonlinear iteration for the linear problems, and cancall the fast nonlinear solver (e.g. Newton method) for the nonlinear problems. Thepositivity, existence and uniqueness of the cell-centered solutions obtained on the corrector phase are analyzed, and the scheme on quasi-uniform meshes is proved to be $L^2$- and $H^1$-stable under some assumptions. Numerical experiments demonstrate theaccuracy, efficiency and positivity of the scheme on various distorted meshes.     

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper, we propose a high-order accurate discontinuous Galerkin(DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state andprovably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitationon rectangular meshes, the extension to triangular meshes is conceptually plausiblebut highly nontrivial. We first introduce a special way to recover the equilibrium stateand then design a group of novel variables at the interface of two adjacent cells, whichplays an important role in the well-balanced and positivity-preserving properties. Onemain challenge is that the well-balanced schemes may not have the weak positivityproperty. In order to achieve the well-balanced and positivity-preserving propertiessimultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivityproperty. A simple existing limiter can be applied to enforce the positivity-preservingproperty, without losing high-order accuracy and conservation. Extensive one- andtwo-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness.     

Discrete Duality Finite Volume Discretization of the Thermal-$P_N$ Radiative Transfer Equations on General Meshes

The discrete duality finite volume method has proven to be a practical toolfor discretizing partial differential equations coming from a wide variety of areas ofphysics on nearly arbitrary meshes. The main ingredients of the method are: (1) useof three meshes, (2) use of the Gauss-Green theorem for the approximation of derivatives, (3) discrete integration by parts. In this article we propose to extend this methodto the coupled grey thermal-$P_N$ radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation. The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit areproved while the experimental second-order accuracy is observed with manufacturedsolutions. Several numerical experiments are reported which show the good behaviorof the method.     

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.     

A New Type of High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes

In this paper, a new type of third-order and fourth-order weighted essentially non-oscillatory (WENO) schemes is designed for simulating the Hamilton-Jacobi equations on triangular meshes. We design such schemes with the use of the nodal information defined on five unequal-sized spatial stencils, the application of monotone Hamiltonians as a building block, the artificial set of positive linear weights to make up high-order approximations in smooth regions simultaneously avoiding spurious oscillations nearby discontinuities of the derivatives of the solutions. The spatial reconstructions are convex combinations of the derivatives of a modified cubic/quartic polynomial defined on a big spatial stencil and four quadratic polynomials defined on small spatial stencils, and a third-order TVD Runge-Kutta method is used for the time discretization. The main advantages of these WENO schemes are their efficiency, simplicity, and can be easily implemented to higher dimensional unstructured meshes. Extensive numerical tests are performed to illustrate the good performance of such new WENO schemes.     

A Nonlinear Finite Volume Scheme Preserving Maximum Principle for Diffusion Equations

In this paper we propose a new nonlinear cell-centered finite volume scheme on general polygonal meshes for two dimensional anisotropic diffusion problems, which preserves discrete maximum principle (DMP). The scheme is based on the so-called diamond scheme with a nonlinear treatment on its tangential flux to obtain a local maximum principle (LMP) structure. It is well-known that existing DMP preserving diffusion schemes suffer from the fact that auxiliary unknowns should be presented as a convex combination of primary unknowns. In this paper, to get rid of this constraint a nonlinearization strategy is introduced and it requires only a second-order accurate approximation for auxiliary unknowns. Numerical results show that this scheme has second-order accuracy, preserves maximum and minimum for solutions and is conservative.     

Arbitrarily High-Order (Weighted) Essentially Non-Oscillatory Finite Difference Schemes for Anelastic Flows on Staggered Meshes

We propose a WENO finite difference scheme to approximate anelastic flows, and scalars advected by them, on staggered grids. In contrast to existing WENO schemes on staggered grids, the proposed scheme is designed to be arbitrarily high-order accurate as it judiciously combines ENO interpolations of velocities with WENO reconstructions of spatial derivatives. A set of numerical experiments are presented to demonstrate the increase in accuracy and robustness with the proposed scheme, when compared to existing WENO schemes and state-of-the-art central finite difference schemes.     

Continuous Finite Element Subgrid Basis Functions for Discontinuous Galerkin Schemes on Unstructured Polygonal Voronoi Meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG)method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classicalpolynomials of degree $N$ inside each element, in our new approach the discrete solutionis represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DGmethod on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on eachsub-triangle are defined, as usual, on a universal reference element, hence allowing tocompute universal mass, flux and stiffness matrices for the subgrid triangles once andfor all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructurednature of the computational grid. High order of accuracy in time is achieved thanksto the ADER approach, making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical resultshave been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtainedby the corresponding modal DG version of the scheme.     

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressibleNavier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite elementmethod for the space discretization. This class of computational solvers benefits fromthe geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows usingtime steps larger than its Eulerian counterparts. In the current study, we implementthree-dimensional limiters to convert the proposed solver to a fully mass-conservativeand essentially monotonicity-preserving method in addition of a low computationalcost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. Theproposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical resultsillustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominatedflow problems on unstructured tetrahedral meshes.